 Solution

For fixed $s$, the factor $n^{-s}$ grows at most polynomially in $n$, whereas $|z{|}^n$ decays exponentially when $|z|<1$. The defining series for the polylogarithm therefore converges absolutely and locally uniformly for every complex $s$ and $|z|<1$. It is holomorphic in $z$ on the unit disc and entire in $s$ there.